Extend A131792 ?

[Martin Fuller]

The following modification is faster and uses less memory in PARI 2.3.2

{T(n,k)=if(n==0,1,polcoeff(prod(j=0,n-1,(1-x^min(2^j+1,k+1))/(1-x)+x*O(x^k)),k))}

I'm also interested in the 'shape' of the rows of A131791.  It looks to
me as though they tend to a limit curve if scaled appropriately, e.g.
scaled to fit a [0,1] box by f_n(x) = T(n,[x*2^n])/A028361(n-1).  In
this setup I think the limit curve f(x) satisfies f(0)=0, f(1-x)=f(x),
f(1/2)=1, f'(x)=2f(2x) for x<=1/2.  Is this is solvable?  There may be
a way forward with Fourier series which I don't have time to explore
today.

Martin Fuller

--- Max Alekseyev <maxale at gmail.com> wrote:

> Paul,
> 
> Your program works perfectly in the current CVS version of PARI/GP
> (development CHANGES-1.1864).
> 
> The following values are obtained in matter of seconds:
> 
> 1, 1, 2, 6, 21, 76, 280, 1045, 3936, 14925,56892, 217791, 836706,
> 3224157, 12456225, 48232162, 187131664, 727309265, 2831193004,
> 11036424667, 43076087806, 168322335246, 658416150496, 2577945422410,
> 10102468839284, 39621592646545, 155510743708961
> 
> Regards,
> Max
> 
> On 8/30/07, Paul D. Hanna <pauldhanna at juno.com> wrote:
> >
> >
> >
> > Seqfans,
> >      Would someone be able to extend A131792, which begins:
> > 1,1,2,6,21,76,280,1045,3936,14925,56892,217791,836706,3224157,
> > 12456225,48232162,
> >
> > I can get only the next term, 187131664,  before I get "degree
> overflow"
> > using PARI (see program at bottom of this message).
> >
> > A131792 is the main diagonal of irregular triangle A131791 (see
> below)
> > where A131791(n,k) = the coefficient of x^k of the row g.f. given
> by:
> >
> > T(n,k) = [x^k] Product_{j=0..n-1} [ (1 - x^(2^j+1) ) / (1-x) ]
> > for n>0, k={0..2^n-1},  with T(0,0)=1.
> >
> > Of course, diagonal n (below the main diagonal) equals the
> convolution
> > of A000108^n (Catalan sequence to n-th power) and this main
> diagonal.
> >
> > Thanks,
> >       Paul
> > -------------------------------------------------------------------
> >
> >
> > Triangle A131791 begins:
> >
> > 1;
> >
> > 1, 1;
> >
> > 1, 2, 2, 1;
> >
> > 1, 3, 5, 6, 6, 5, 3, 1;
> >
> > 1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1; ...
> >
> > 1, 5, 14, 29, 50, 76, 105, 135, 165, 194, 220, 241, 256, 265, 269,
> 270,
> >
> >  270, 269, 265, 256, 241, 220, 194, 165, 135, 105, 76, 50, 29, 14,
> 5, 1; ...
> >
> >
> >
> > PROGRAM (PARI):
> >
> >
>
{T(n,k)=if(n==0,1,polcoeff(prod(j=0,n-1,(1-x^(2^j+1))/(1-x)+x*O(x^k)),k))}
> >
> > /* Print Triangle A131791: */
> >
> > for(n=0,8,for(k=0,2^n-1,print1(T(n,k),","));print(""))
> >
> > /* Print Main Diagonal A131792: */
> >
> > for(n=0,36,print1(T(n,n),","))
> >
> > END.
>